Background Results Example

Torsion Divisor Classes

Sandra Spiroff Joint with Sean Sather-Wagstaff, North Dakota State Univ.

University of Mississippi

AMS Fall Eastern Sectional Meeting Halifax, Nova Scotia, October 18, 2014

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Outline

Background

Results

Example

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Background

Let A → B be a homomorphism of Noetherian normal integral domains. Under certain circumstances, one can show that such a map induces a group homomorphism Cl(A) → Cl(B). • (faithfully) flat ring homomorphisms; The map Cl(A) → Cl(B) is given by [a] 7→ [a ⊗A B]. • integral extensions;

• (Danilov, 1970) the natural surjection A[[T ]] → A;

• (Lipman, 1979) any surjection of the form A → A/tA, assuming that A and A/tA are both Noetherian normal integral domains.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

• (S-W, S, 2009) homomorphisms of finite flat dimension between normal domains. e.g., A → A/I , where I is a prime ideal of A with finite projective dimension, A and A/I are normal domains. BB The map Cl(A) → Cl(B) is given by [a] 7→ [(a ⊗A B) ].

• (S-W, S, 2013) A → (A/I )0, where A is a normal domain, I a prime ideal of A such that A/I is excellent and satisfies (R1), and (A/I )0 the integral closure of A/I .

The kernel of this induced map has been studied by V.I. Danilov, P. Griffith, J. Lipman, C. Miller, D. Weston, and others.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Some Definitions A The dual of an A-module a is a := HomA(a, A).

AA There is a map σ : a → a , defined by σ(x) = ϕx , where ϕx (f ) := f (x), for f ∈ HomA(a, A). If σ is an isomorphism, then a is said to be reflexive.

The divisor class group of A, denoted Cl(A), is the group of isomorphism classes of reflexive ideals of A, or equivalently, reflexive A-modules of rank one. An element [a] in Cl(A) is called a divisor class.

AA • multiplication is defined by [a] · [b] = [(a ⊗A b) ]; • the identity element is [A];

−1 A • [a] = [a ] = HomA(a, A).

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

FACT: A Noetherian integral domain is a unique factorization domain if and only if every height one prime is principal.

Example

∼ • Let A = C[X ,Y ,Z,W ]/(XY −ZW). Then Cl(A) = Z. 2 ∼ • Let B = C[X ,Y ,Z]/(XY −Z ). Then Cl(B) = Z/2Z.

There is a map Cl(A) → Cl(B) induced from A →→ B.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Theorem (Griffith, Weston 1994*) Let (A, m) be an excellent, local, normal domain and let t be a principal prime element in m such that A/tA satisfies the condition (R1). Let e > 1 be an integer which represents a unit in A. Then the kernel of the homomorphism Cl(A) → Cl((A/tA)0) contains no element of order e.

BB Recall: The map Cl(A) → Cl(B) is given by [a] 7→ [(a ⊗A B) ], where B = (A/I )0.

*P. Griffith and D. Weston, Restrictions of torsion divisor classes to hypersurfaces, J. Algebra 167 (1994), no. 2, 473–487.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Outline of their proof

• Suppose [a] is an element of order e in kernel of Cl(A) → Cl((A/tA)0); let α ∈ A be such that a(e) = αA.

• They describe an A-algebra structure on the direct sum R = A ⊕ a ⊕ a(2) ⊕ · · · ⊕ a(e−1) that makes R isomorphic to the integral closure of A[T ]/(T e − α).

0 • They then show that the integral closure of R ⊗A (A/tA) is ´etaleover (A/tA)0 and conclude that e = 1.

0 • Subtle Detail: That the integral closure of R ⊗A (A/tA) is local.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

The best way to be certain of this claim is to use a result of M. Hochster and C. Huneke, interpreting the integral closure of 0 0 R ⊗A (A/tA) as the S2-ification of R ⊗A (A/tA) .

M. Hochster and C. Huneke, Indecomposable canonical modules and connectedness, in: Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra, South Hadley, MA, 1992, AMS, 1994, pp. 197-208.

Fact (Hochster, Huneke)

Let (B, n) be an excellent equidimensional local ring. If B is (S2) and x1,..., xk is a part of a system of parameters, then the S2-ification of B/(x1,..., xk )B is local.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Definition For a Noetherian integral domain B, a subring S of the total ring of quotients of B is an S2-ification of B if: • S is module finite over B;

• S satisfies the Serre condition (S2) over B; and • Coker(B → S) has no prime ideal of B of height less than two in its support.

Recall: S satisfies the (S2) condition over B if for every prime ideal p of B, depth Sp ≥ min{2, ht p}.

FACT: If A is a local Noetherian normal domain that satisfies (R1), then the S2-ification S of A coincides with the integral closure of A in its field of fractions.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Theorem One [Sather-Wagstaff, S-] Let (A, m) be an excellent, local, normal domain and I a prime ideal of A with finite projective dimension such that: (i) A = A/I is normal; (ii) µ(I ) ≤ dim A − 2, where µ(I ) is the minimal number of generators of I ; and (iii) I is a complete intersection on the punctured spectrum of A, i.e., for each prime ideal p 6= m, the localization Ip is either equal to Ap or generated by a regular sequence in Ap. Then for any integer e > 1 which represents a unit in A, the kernel of the homomorphism Cl(A) → Cl(A) contains no elements of order e.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Fact (Hochster, Huneke 1992**) For (T , n) a complete local equidimensional ring such that (0) has no embedded primes, the following conditions are equivalent: dim T • Hn (T ) is indecomposable; • The canonical module of T is indecomposable;

• The S2-ification of T is local; • For every ideal J of height at least two, Spec(T ) − V (J) is connected; • Given any two distinct minimal primes p, q of T , there is a sequence of minimal primes p = p0,..., pr = q such that for 0 ≤ i < r, ht(pi + pi+1) ≤ 1.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

A significant tool in the proof of theorem one is the next result.

Theorem Two [Sather-Wagstaff, S-] Let (A, m) be a complete, local, normal domain such that A/m is separably closed and let I be a prime ideal of A with finite projective dimension such that A = A/I is normal. Let e > 1 be an integer which represents a unit in A and assume that A contains a primitive e-th root of unity. Let [a] be an element in Cl(A) with order e. Set R = A ⊕ a ⊕ a(2) ⊕ · · · ⊕ a(e−1). If any of the five equivalent conditions of the HH Fact hold for R/IR, then [a] ∈/ Ker(Cl(A) → Cl(A)).

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Lemma: Let P ∈ Spec(R) and set p = P ∩ A. Then ht p = ht P. Furthermore, if Ap is regular, then the extensions Ap → Rp and Ap → RP are ´etale.In particular, the extension A ,→ R is ´etalein codimension one.

Definition We say that R is ´etalein codimension one (or more accurately, in codimension less than or equal to one) over A if, for every prime ideal P of S with height less than or equal to one and p = P ∩ A, the ring RP is ´etaleover Ap.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Lemma:The ring R is a complete local normal domain.

Lemma: Set R¯ = R ⊗A A¯ = R/IR. The ring R¯ is equidimensional, complete, and local. In particular, Fact HH applies with T = R¯.

Lemma: The extension A¯ → R¯ is ´etalein codimension one. Moreover, the ring R¯ satisfies the Serre condition (R1).

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Lemma: For σ : R¯ → R¯ A¯A¯ , we have Ker σ = j(R¯).

Definition Denote by j(R¯) the largest ideal which is a submodule of R¯ of dimension smaller than dim R¯. Specifically, ¯ ¯ ¯ ¯ j(R) = {¯r ∈ R : dim(R/ annR¯ (¯r)) < dim R}.

Lemma: The ring R¯/j(R¯) satisfies (R1). Its integral closure 0 (R¯/j(R¯)) in its total ring of quotients is its S2-ification.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Lemma: The A¯-module R¯ A¯A¯ is free of rank e and thus satisfies (S2) over A¯.

0 Lemma:(R¯/j(R¯)) satisfies (S2) as an A¯-module and as a ring.

¯ ¯ ¯ ¯ Lemma:(R¯/j(R¯))0 =∼ (R¯/j(R¯))AA =∼ R¯ AA as A¯-isomorphisms.

Remark 0 Upshot: (R¯/j(R¯)) is the S2-ification of R¯.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Lemma: The ring (R¯/j(R¯))0 is a normal domain.

Lemma: The map A¯ → (R¯/j(R¯))0 is ´etalein codimension one.

Lemma: The extension A¯ ,→ (R¯/j(R¯))0 is ´etale.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Conclusion of proof of Theorem Two.

Let S denote (R¯/j(R¯))0. The extension A¯ ,→ S is ´etaleand local and S is free over A¯ of rank e. There are isomorphisms

A¯/m¯ =∼ S/m¯ S =∼ (A¯/m¯ )e .

It follows that e = 1. Contradiction.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Theorem One Hypotheses:

(A, m) is an excellent, local, normal domain and I a prime ideal of A with finite projective dimension such that: (i) A = A/I is normal; (ii) µ(I ) ≤ dim A − 2, where µ(I ) is the minimal number of generators of I ; and (iii) I is a complete intersection on the punctured spectrum of A, i.e., for each prime ideal p 6= m, the localization Ip is either equal to Ap or generated by a regular sequence in Ap.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Example Let (A, m) be an excellent normal local integral domain, and let the positive integer e represent a unit in A. Assume that A has an A-regular sequence f1,..., f6 ∈ m such that A/(f1,..., f6)A is a normal domain. (Examples of such rings are constructed in S-W, S 2009.) In particular, we have 6 ≤ depth A ≤ dim A.

Localize at a minimal prime of the ideal J = (f1,..., f6)A, if necessary, to assume that J is m-primary. It follows that A is Cohen-Macaulay of dimension 6. Arrange the sequence f ,..., f in   1 6 a 2 × 3 matrix F = f1 f2 f3 , and consider the ideal I = I (F ) f4 f5 f6 2 generated by the 2 × 2 minors d1 = f1f5 − f2f4, d2 = f2f6 − f3f5, d3 = f1f6 − f3f4.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Thank You, eh.

Merci, eh.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Let A be a ring, and let R be an A-algebra. Definition For a prime ideal P of R and p = P ∩ A, we say that P is unramified over A if pRP = PRP and RP /pRP is a separable field extension of Ap/pAp.

Definition We say that R is unramified in codimension i (or more accurately, in codimension less than or equal to i) over A if every prime ideal of R with height less than or equal to i is unramified over A.

Definition For R to be unramified over A means that (1) every prime ideal of R is unramified over A, and (2) for every p ∈ Spec(A) there are only finitely many prime ideals of R lying over p.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

• A satisfies the hypotheses of the first Theorem (excellent, local, normal domain);

• I is a height-2 prime ideal of A of finite projective dimension generated by 3 elements (so µ(I ) < dim A − 2) such that A/I is a normal domain;

• I is a complete intersection on the punctured spectrum; i.e., ◦ for p ∈ Spec (A) such that I ⊆ p, Ip is generated by an Ap-regular sequence.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

(n1) (nr ) • a = p1 ∩ · · · ∩ pr , pi height one primes in A

(e) (en1) (enr ) • a = p1 ∩ · · · ∩ pr = αA and vpi (a) = eni

• ∃b ∈ A such that vpi (b) = ni for i = 1,..., r, else vq(b) ≥ 0

e • Set u = b /a ∈ K. Then vpi (u) = eni − eni = 0 and e vq(u) = vq(b ) ≥ 0. √ • Let e u be a fixed e-th root of u in an algebraic closure of K. √ √ Then e a := b/ e u is an e-th root of a. Also, au = be .

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

The ring √ " √ # " √ #  e u  e u2 e ue−1 R = A ⊕ a · ⊕ a(2) · ⊕ · · · ⊕ a(e−1) · b b2 be−1 √ √ is the integral closure of A in the field extension K[ e a] = K[ e u]; i.e., R is a domain.

(i) (j) The ring structure on R: for ai ∈ a and aj ∈ a , we have

 √e ui+j √e ! √e !  ui uj ai aj if i + j < e a a = √bi+j √ i bi j bj e ui+j a a e ui+j−e a a = i j if i + j ≥ e.  i j bi+j a bi+j−e

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Key References

1. P. Griffith, Some results in local rings on ramification in low codimension, J. Algebra 137 (1991), no. 2, 473–490. 2. P. Griffith and D. Weston, Restrictions of torsion divisor classes to hypersurfaces, J. Algebra 167 (1994), no. 2, 473–487. 3. M. Hochster and C. Huneke, Indecomposable canonical modules and connectedness, in: Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra, South Hadley, MA, 1992, AMS, 1994, pp. 197-208. 4. S. Sather-Wagstaff and S. Spiroff, Maps on divisor class groups induced by ring homomorphisms of finite flat dimension, J. Comm. Algebra 1 (2009), no. 3, 567–589.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Proposition Let C ,→ D be a module finite extension of local normal domains which is unramified in codimension one. If t is an element in the maximal ideal of C such that C¯ = C/tC satisfies the condition (R1), then D¯ = D/tD satisfies the condition (R1) as well, and C¯ ,→ D¯ is unramified in codimension one.

S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example

Corollary Under the same hypotheses as in the above Proposition, if the rings are complete, then the integral closures (C¯)0 and (D¯ )0 are local integral domains.

Corollary Under the same hypotheses as in the previous corollary, the extension (C¯)0 → (D¯ )0 is module finite and unramified in codimension one.

S. Spiroff, University of Mississippi Torsion Divisor Classes